Two Crossing-Cubic Vector Fields

In this chapter, a cubic dynamical system possessing two crossing-variable cubic vector fields are discussed. The appearing and switching bifurcations for centers and saddles of equilibriums are discussed, and the limit cycle and homoclinic networks are presented through the first integral manifolds. The corresponding equilibriums include simple center and saddles; parabola-saddle; and third-order saddles and centers, (2,2)-double-diagonal inflection saddles, (3,2)-parabola-saddle, and (3,3)-double-third-order saddles and centers. The double third-order saddle and centers are the appearing bifurcations of the 3×3 network of saddles and centers with homoclinic orbits. The double-third-order saddles and centers are also the switching bifurcations of parabola-saddle with saddles and centers. The double-third-order saddles and centers are also the switching bifurcation for the networks of center (saddle), parabola-saddles, and inflection saddles. The networks of saddle and center for such nonlinear systems with two cubic variable-crossing vector fields are presented.